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arXiv:2204.10790 · Physical review. D/Physical review. D.

Analogous Hawking radiation and quantum entanglement in two-component Bose-Einstein condensates: The gapped excitations

Wei-Can Syu, Da-Shin Lee, Chi-Yong Lin

Published Aug 5, 2022• 10 citations
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Abstract

The condensates of cold atoms at zero temperature in the tunable binary Bose-Einstein condensate system are studied with the Rabi transition between atomic hyperfine states where the system can be represented by a coupled two-field model of gapless excitations and gapped excitations. We set up the configuration of the supersonic and subsonic regimes with the acoustic horizon between them in the elongated two-component Bose-Einstein condensates, trying to mimic Hawking radiations, in particular due to the gapped excitations. The simplified steplike sound speed change is adopted for the subsonic-supersonic transition so that the model can be analytically treatable. The effective-energy gap term in the dispersion relation of the gapped excitations introduces the threshold frequency ${\ensuremath{\omega}}_{\mathrm{min}}$ in the subsonic regime, below which the propagating modes do not exist. Thus, the particle spectrum of the Hawking modes significantly deviates from that of the gapless cases near the threshold frequency due to the modified gray-body factor, which vanishes as the mode frequency is below ${\ensuremath{\omega}}_{\mathrm{min}}$. The influence from the gapped excitations to the quantum entanglement of the Hawking mode and its partner of the gapless excitations is also studied according to the Peres-Horodecki-Simon (PHS) criterion. It is found that the presence of the gapped excitations will deteriorate the quantumness of the pair modes of the gapless excitations when the frequency of the pair modes in particular is around $\ensuremath{\omega}\ensuremath{\sim}{\ensuremath{\omega}}_{\mathrm{min}}$. On top of that, when the coupling constant between the gapless and gapped excitations becomes large enough, the huge particle density of the gapped excitations in the small $\ensuremath{\omega}$ regime will significantly disentangle the pair modes of the gapless excitations. The detailed time-dependent PHS criterion will be discussed.

DOI
10.1103/physrevd.106.044016

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The condensates of cold atoms at zero temperature in the tunable binary Bose-Einstein condensate system are studied with the Rabi transition between atomic hyperfine states where the system can be represented by a coupled two-field model of gapless excitations and gapped excitations. We set up the configuration of the supersonic and subsonic regimes with the acoustic horizon between them in the elongated two-component Bose-Einstein condensates, trying to mimic Hawking radiations, in particular due to the gapped excitations. The simplified steplike sound speed change is adopted for the subsonic-supersonic transition so that the model can be analytically treatable. The effective-energy gap term in the dispersion relation of the gapped excitations introduces the threshold frequency \ensuremathωmin{\ensuremath{\omega}}_{\mathrm{min}}\ensuremathωmin​ in the subsonic regime, below which the propagating modes do not exist. Thus, the particle spectrum of the Hawking modes significantly deviates from that of the gapless cases near the threshold frequency due to the modified gray-body factor, which vanishes as the mode frequency is below \ensuremathωmin{\ensuremath{\omega}}_{\mathrm{min}}\ensuremathωmin​. The influence from the gapped excitations to the quantum entanglement of the Hawking mode and its partner of the gapless excitations is also studied according to the Peres-Horodecki-Simon (PHS) criterion. It is found that the presence of the gapped excitations will deteriorate the quantumness of the pair modes of the gapless excitations when the frequency of the pair modes in particular is around \ensuremathω\ensuremath∼\ensuremathωmin\ensuremath{\omega}\ensuremath{\sim}{\ensuremath{\omega}}_{\mathrm{min}}\ensuremathω\ensuremath∼\ensuremathωmin​. On top of that, when the coupling constant between the gapless and gapped excitations becomes large enough, the huge particle density of the gapped excitations in the small \ensuremathω\ensuremath{\omega}\ensuremathω regime will significantly disentangle the pair modes of the gapless excitations. The detailed time-dependent PHS criterion will be discussed.
We have studied analytically the approximate solutions to the gapped mode equations in the hydrodynamic regime for a class of binary Bose-Einstein condensate acoustic black holes. The horizon from the transonic flow is formed by manipulating the phonon sound speed and the flow velocity with the experimentally accessible parameters. The asymptotic modes of various scattering processes are constructed from which to obtain scattering coefficients and then to further decompose the field operator in terms of the asymptotic states. Also, the Unruh state is introduced to be the appropriate state for the description of gravitational collapse of the black hole. The particle densities of the outgoing modes are computed. The effective energy gap term in the dispersion relation of the gapped excitations introduces the threshold frequency ${\ensuremath{\omega}}_{r}$ in the subsonic regime, below which the propagating modes do not exist. Thus, the particle spectrum of the analogous Hawking modes in the exterior of the horizon of the subsonic region significantly deviates from that of the gapless cases near the threshold frequency due to the modified graybody factor, which vanishes as the mode frequency is below ${\ensuremath{\omega}}_{r}$. However, in the interior region of the horizon of the supersonic region, the spectrum of the particle production of the Hawking partner has the nonthermal feature. The correlators between the analog Hawking mode and its partner of relevance to the experimental observations are also investigated and show some peaks near the threshold frequency ${\ensuremath{\omega}}_{r}$ resulting from the gap energy term to be seen in future experiments.
We investigate the properties of the condensates of cold atoms at zero temperature in the tunable binary Bose-Einstein condensate system with a Rabi transition between atomic hyperfine states. We use this system to examine the effect of quantum fluctuations in a tunable quantum gas on phonon propagation. We show that the system can be represented by a coupled two-field model of a gapless phonon and a gapped mode, which are analogous to the Goldstone and Higgs particles in particle physics. We then further trace out the gapped modes to give an effective purely phononic theory using closed-time-path formalism. In particular, we are interested in the sound cone fluctuations due to the variation of the speed-of-sound acoustic metric, induced by quantum fluctuations of the gapped modes. These fluctuations can be interpreted as inducing a stochastic space-time, and thus are regarded as analogue phenomena of light cone fluctuations presumably arising from quantum gravity effects. The effects of fluctuations can be displayed in the variation in the travel time of sound waves. We suggest the relevant experiments to discuss the possibility of experimental observations.
We analytically study the low-frequency properties of the analogue Hawking effect in Bose-Einstein condensates. We show that in one-dimensional flows displaying an analogue horizon, the Hawking effect is dominant in the low-frequency regime. This happens despite nonvanishing grey-body factors, that is, the coupling of the Hawking mode and its partner to the mode propagating with the flow. To show this, we obtained analytical expressions for the scattering coefficients, in general flows and taking into account the full Bogoliubov dispersion relation. We discuss the obtained expressions for the grey-body factors. In particular, we show that they can be significantly decreased if the flow obeys a conformal coupling condition. We argue that in the presence of a small but non-zero temperature, reducing grey-body factors greatly facilitates the observation of entanglement, that is, establishing that the state of the Hawking mode and its partner is non-separable.
We investigate the Beliaev damping of gapped excitations in a homogenous two-component Bose-Einstein condensate with Rabi coupling and find different damping behavior of gapped excitations in two quantum phases. In the paramagnetic phase, due to constraints imposed by the Z2{\mathbb{Z}}_{2}Z2​ symmetry and energy-momentum conservation, the only damping channel available, that of decaying into one phonon and one new gapped mode, cannot occur unless the group velocity of the gapped modes surpasses the sound speed. Conversely, in the ferromagnetic phase, three distinct decay channels exist. Once the momentum of the excitations falls below a critical threshold, only one of these channels, that of decaying into two gapless modes, is activated, and the damping rate remains finite even at zero momentum.
We theoretically study stimulated and spontaneous Hawking emission from an analog horizon for spin modes in a two-component Bose–Einstein condensate, both with and without a coherent coupling between the two components. We highlight the conceptual and practical advantages that these systems offer to the experimental observation of the phenomenon, namely the possibility of working with massive elementary excitations and the experimental accessibility of the different quadratures of the spin excitations. In particular, we go beyond the relativistic regimes previously addressed in the literature, and identify various observables that show a signature of the Hawking process, as well as additional features associated with the massive nature of the modes, such as undulations . Semi-analytical calculations of the scattering properties of the horizon and of two-point correlation functions of the emitted radiation in an ideal stationary setup are supported by time-dependent numerical simulations based on Gross–Pitaevskii and Bogoliubov theory.
Observing quantum particle creation by black holes (Hawking radiation) in the astrophysical context is, in ordinary situations, hopeless. Nevertheless the Hawking effect, which depends only on kinematical properties of wave propagation in the presence of horizons, is present also in nongravitational contexts, for instance in stationary fluids undergoing supersonic flow. We present results on how to observe the analog Hawking radiation in Bose-Einstein condensates by a direct measurement of the density correlations due to the phonon pairs (Hawking quanta-partner) created by the acoustic horizon.